Uncertainty Quantification in Fluid-Structure Interaction Simulations Using a Simplex Elements Stochastic Collocation Approach

Abstract

Numerical errors in multi-physics simulations start to reach acceptable engineering accuracy levels due to the increasing availability of computational resources. Nowadays, uncertainties in modeling multi-scale phenomena in fluid-structure, fluid-thermal, and aero-acoustic interactions have a larger effect on the accuracy of computational predictions than discretization errors. It is therefore especially important in multi-physics applications to systematically quantify the effect of physical variations on a routine basis. Furthermore, unsteady fluid-structure interaction applications are practical aeronautical examples of dynamical systems which are known to amplify initial variations with time. In these problems, natural irreducible input variability can trigger the earlier onset of unstable flutter behavior, which can lead to unexpected fatigue damage and structural failure. Polynomial Chaos uncertainty quantification methods 2, 7, 8, 12, 13, 17, 20, 21, 29 however usually result in a fast increasing number of samples with time to resolve the effect of random parameters in dynamical systems with a constant accuracy. Resolving the asymptotic stochastic effect, which is of practical interest in postflutter analysis, can in these long time integration problems lead to thousands of required samples. The increasing number of samples is caused by the increasing nonlinearity of the response surface for increasing integration times. This effect is especially profound in problems with oscillatory solutions in which the frequency of the response is affected by the random parameters. The frequency differences between the realizations lead to increasing phase differences with time, which in turn result in an increasingly oscillatory response surface and more required samples. In order to enable efficient uncertainty quantification in time-dependent simulations, a special uncertainty quantification methodology for unsteady oscillatory problems is developed. The approach based on time-independent parameterization of oscillatory samples achieves a constant uncertainty quantification interpolation accuracy in time with a constant number of samples. A parameterization in terms of the time-independent functionals frequency, relative phase, amplitude, a reference value, and the normalized period shape is used for period-1 responses. The extension with a damping factor and an algorithm for identifying higher-period shape functions is also applicable to more complex and non-periodic realizations. A second uncertainty quantification formulation for achieving a constant accuracy in time with a constant number of samples is developed based on interpolation of oscillatory samples at constant phase instead of at constant time. The scaling of the samples with their phase eliminates the effect of the increasing phase